Lallart M. (ed.) Ferroelectrics

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Thermal Conduction Across Ferroelectric Phase Transitions: Results on Selected Systems

159

pyroelectric detector, supported on a thermally thick conductive backing. Since the PPE

signal depends on properties of the detector which are also temperature dependent, an

accurate temperature calibration of the system must be carried out. The advantage of a

thermally thick backing is that there will be sufficient heat exchange between the heated

pyroelectric detector and the backing, so that signal fluctuations are reduced to a minimum.

This method can, in principle, be adapted to all temperature ranges for all samples and is

not limited by the thermal properties of the sample.

The PPE effect is based on the use of a pyroelectric transducer to detect the temperature rise

due to periodic heating of a sample by induced light. The temperature variations in the

detector give rise to an electrical current, which is proportional to the rate of change of the

average heat content, given by (Mandelis & Zver, 1985)

()

iPA













(2)

where P is the pyroelectric coefficient of the detector, A is the area of the detector and

() ( , )

txtdx



 







(3)

is the spatially averaged temperature variation over the thickness of the detector, L

For a thermally thick sample with



 L

, and a thermally thick pyroelectric detector with



 L

, where μ

and μ

are the thermal diffusion lengths of the sample and detector

respectively, the expressions for the PPE amplitude and phase give expressions for the

values of the thermal diffusivity

and effusivity e

which allow a simultaneous

determination of the thermal conductivity and heat capacity if the density



of the sample is

known. The expressions for the temperature dependent PPE amplitude and phase under the

above conditions are given by (Menon & Philip, 2000)

exp

()

(,)

() ()

()

os d

dpd

IAR PT

VfT

Tc T





































 





























(4a)

( , ) tan ( )

()







 







(4b)

where T is the temperature and c

and



are the heat capacity (at constant pressure) and

density of the detector respectively. f

is the characteristic frequency at which the sample

goes from a thermally thin to thermally thick regime. From these two expressions it is clear

that the thermal diffusivity



of the sample can be calculated from the phase of the PPE

signal, which when substituted into the expression for the PPE amplitude, gives the thermal

effusivity of the sample. From these the thermal conductivity and heat capacity of the

sample can be calculated from the following relations:

() () ()

sss

TeT T  (5a)

Ferroelectrics – Physical Effects

160

()

() ()





(5b)

A temperature calibration of the PPE detector is necessary here as all the parameters in

equations (4a) and (4b) are temperature dependent. All the thermal parameters can be

calculated as functions of the sample temperature, provided that the temperature

dependences of the parameters of the pyroelectric detector are known.

4. Experimental methods in PPE measurements

A sample set-up of the type shown in Fig. 1 is generally used for these measurements

(Menon & Philip, 2000). A 120 mW He - Cd laser of λ = 442 nm, modulated by a mechanical

chopper, has been used as the optical heating source. A 28 μm thick film of PVDF with

pyroelectric coefficient P = 0.25 ×10

−8

V cm

−1

at room temperature has been used as the

pyroelectric detector. The sample is attached to the pyroelectric detector with a thermally

very thin layer of a heat sink compound whose contribution to the signal is negligible. The

pyroelectric detector attached to the sample is placed on a thermally thick backing medium

(copper) which satisfies the boundary condition specified above. The frequency of

modulation of the light is kept high enough to ensure that the PVDF film, the sample and

the backing medium are all thermally thick. The signal output is measured with a lock-in

amplifier. The sample-detector-backing assembly is enclosed in a chamber whose

temperature can be varied and controlled as desired. Measurements as a function of

temperature have been made at a low heating rate with special care near transition points. A

block diagram of the experimental set up is shown in Fig. 2 for illustration.

The experimental set up and procedure should be calibrated and tested to ensure that even

minor variations in heat capacity and thermal conductivity do get reflected in the

measurements. Practically one measures the PPE amplitude and phase as function of

modulation frequency, limiting the frequency to low values so that the sample, detector and

backing are all thermally thick. From the amplitude and phase variations one can determine

the thermal effusivity and thermal diffusivity following equations (4a) and (4b) respectively.

From the values of thermal diffusivity and thermal effusivity, the values of thermal

conductivity and specific heat capacity can be determined following equations (5a) and (5b).

Fig. 1. The sample configuration for the photopyroelectric set-up

Thermal Conduction Across Ferroelectric Phase Transitions: Results on Selected Systems

161

Fig. 2.

Block diagram of the experimental set-up used for PPE measurements

Practically one measures the photopyroelectric signal amplitude and phase as function of

modulation frequency. One will have inverse frequency dependence for the amplitude and

phase beyond the critical frequency when the boundary conditions assumed are satisfied. A

fitting of the variations of PPE amplitude and phase with the relations connecting thermal

diffusivity and effusivity with phase and amplitude respectively enables one to determine

the thermal diffusivity and effusivity. Typical variations of PPE amplitude and phase with

modulation frequency obtained during PPE measurements in K

SeO

are shown in figures

3a and 3b respectively. The peaks in the curves correspond to characteristic modulation

frequency for the sample.

5. Results on specific systems and discussion

The variations in the thermal properties of the ferroelectric crystal Triglycine sulphate (TGS)

were reported by Menon & Philip (2000). TGS crystals undergo a para–ferroelectric phase

transition at 49.4

C. This crystal has a monoclinic structure at room temperature. Platelets of

the crystal of sub-millimeter thickness were cut with faces normal to a, b and c axes so that

the direction of propagation of thermal waves was along one of the axes. A very thin layer

of carbon black was coated onto the illuminated surface of the sample to enhance its optical

absorption. Measurements were carried out as a function of temperature from room

temperature (26

C) to 55

C. The thermal thickness of the sample in these experiments was

verified by plotting the PPE amplitude and phase against modulation frequency at a

number of temperatures between room temperature and 55

C. The variations of the PPE

amplitude and phase as functions of temperature were measured keeping the modulation

frequency fixed at 40 Hz. From these, the thermal diffusivity (α

) and effusivity (e

) along the

b axis of TGS were determined as functions of temperature. These are shown in Fig. 4a. The

temperature variations of λ

along the b axis and c

were also determined (shown in Fig. 4b).

The heat capacity results presented in Fig. 4b agree with those already reported in the

literature (Strukov & Levanyuk 1998). Their results showed that thermal conductivity along

the b axis has a minimum value at the transition point. Measurements of the thermal

diffusivity or conductivity along a- and c- axes did not reveal any anomaly at the phase

transition temperature. This was in agreement with thermal diffusivity measurements along

these axes reported earlier (Gaffar et al., 1987).

Ferroelectrics – Physical Effects

162

0 100 200 300 400

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

PPE Amplitude (mV)

Modulation Frequency (Hz)

a-axis

b-axis

c-axis

Fig. 3. (a)

Frequency dependence of the photo-pyroelectric amplitudes along the three

principal axes of K

SeO

at room temperature (Philip & Manjusha, 2009)

0 100 200 300 400

100

120

140

160

180

200

220

240

260

PPE Phase (Degrees)

Modulation Frequency (Hz)

a-axis

b-axis

c-axis

Fig. 3. (b) Frequency dependence of the photopyroelectric phases along the three principal

axes of K

SeO

at room temperature (Philip & Manjusha, 2009)

Ferroelectric crystals, which exhibit incommensurate phase transitions include ammonium

fluroberrylate (Iizumi & Gesi, 1977), potassium selenate (Iizumi et al., 1977), sodium nitrite

(Yamada et al., 1963), thiourea (Goldsmith & White, 1959) etc. Thiourea, with the chemical

formula SC (NH

)

, undergoes successive phase transitions at 169 K (T

), 176 K (T

), 180 K

) and 202 K (T

). Among the five phases (called I, II, III, IV and V) in the order of

increasing temperature, two of them (I and III) are ferroelectric and a superlattice structure

appears in the II, III and IV phases (Elcombe & Tayler, 1968). The crystal structure in the

Thermal Conduction Across Ferroelectric Phase Transitions: Results on Selected Systems

163

room temperature phase V above T

is orthorhombic and belongs to the space group D

Pbnm with four molecules per unit cell (Z = 4). The phase I below T

is a ferroelectric one

having spontaneous polarization along the b-axis, whose crystal structure is also

orthorhombic with four molecules per unit cell (Z = 4).

In the three intermediate phases, II, III and IV between T

and T

, Shiozaki (1971) analyzed

X-ray reflection spectra and concluded that the crystal has an in-commensurate structure.

According to his analysis, just above T

the crystal has a superstructure along the c-axis with

a period about eight times as large as that of phase IV. The period of the super structure

increases as temperature decreases. So in the vicinity of T

, the period is about ten times as

large and at T

the crystal transforms to the ferroelectric phase I, where the period of the

unit cell of the prototype is restored. More elaborate descriptions of the properties of

thiourea are available in literature (Wada et al., 1978, Moudden et al., 1978, Mc Kenzie,

1975a, Mc Kenzie, 1975b, Delahaigue et al., 1975, Chapelle & Benoit, 1977).

The thermal properties described above during the incommensurate-commensurate phase

transition in thiourea were measured employing PPE technique (Menon & Philip, 2003).

Measurements have been done along the three principal directions of thiourea and the

observed anisotropy in thermal transport is discussed. The crystals were cut with their faces

normal to the [100], [010] and [001] directions of the crystallographic a-, b- and c-axes

respectively. Measurements were carried out illuminating the three cut sample faces so that

the propagation of the thermal wave is along one of the symmetry directions. The variations

of thermal conductivity and heat capacity as functions of temperature across the transition

temperatures were measured as outlined above.

Fig. 4. (a)

Variations of the thermal diffusivity (inverted triangles) and thermal effusivity

(triangles) with temperature for TGS along the b axis (Menon & Philip, 2000)

It was seen that both PPE amplitude and phase clearly reflect the three successive phase

transitions in thiourea. The maximum anomaly was at T

, the temperature at which

transition to an in-commensurate phase took place. Anomalies were measured along a, b

Ferroelectrics – Physical Effects

164

Fig. 4. (b)

Variations of the thermal conductivity (inverted triangles) and heat capacity

(triangles) with temperature along the b-axis (Menon & Philip, 2000)

and c directions at the same temperatures. The maximum variation was seen along the b-

direction, the direction in which the crystal possesses spontaneous polarization in the

ferroelectric phase. Fig. 5a shows the variations of thermal diffusivity and thermal effusivity

with temperature along the b-axis of thiourea single crystal. As can be seen in this figure,

thermal diffusivity shows a decrease with temperature, with distinct minima at the three

phase transition points at T

≈169 K, T

≈176K and T

≈ 202 K, in agreement with the already

reported values of transition temperatures. Thermal effusivity exhibits an inverse behaviour.

It increases with temperature, with sharp peaks occurring at the transition temperatures.

Taking into account the various uncertainties of the measurement, the overall uncertainty in

the values of α and e are estimated to be less than 5%. Similar anomalies with smaller

magnitudes have been obtained for the a- and c-directions as well.

Figure 5b shows the variation of heat capacity of thiourea with temperature. As can be seen

in this figure, the three transitions get clearly reflected in the temperature variation of heat

capacity as clear anomalies at the transition points. These heat capacity values agree with

the values reported by earlier workers (Hellwege & Hellwege, 1969). As can be seen, there is

no direction dependence for heat capacity. Figure 5c shows the temperature variation of

thermal conductivity along the three symmetry directions (a, b and c) of thiourea. The

thermal conductivity exhibits significant anisotropy, as is evident from Fig. 5c. The three

transitions get clearly reflected in the thermal conductivity variations as well. The maximum

anomaly at the transition temperatures is seen along the b-axis. The maximum thermal

conduction occurs in the direction of predominant covalent bonding, which is along the b-

axis in thiourea. This is the direction of spontaneous polarization in this crystal.

Dicalcium Lead Propionate (DLP, with chemical formula Ca

Pb (C

COO)

, belonging to

the family of double propionates, is ferroelectric below 333K along the c- axis (Nakamura et

al 1965). It undergoes a para to ferro electric phase transition at 333K (Tc

), which is a second

Thermal Conduction Across Ferroelectric Phase Transitions: Results on Selected Systems

165

Fig. 5. (a)

Temperature variations of thermal diffusivity and thermal effusivity along the b-

axis of thiourea single crystal. Similar variations to a lesser extend were exhibited by a- and

c-directions (Menon & Philip, 2003)

Fig. 5. (b)

Temperature variation of heat capacity along three principal directions of thiourea

single crystal. The inset shows the variation of heat capacity between 160 K and 220 K along

the b-axis (Menon & Philip, 2003)

order one. Upon decreasing the temperature further, it undergoes another phase transition

at 191K (Tc

), which is first order. The transition at Tc

is associated with the movement of

the ethyl group (C

) (Nakamura et al., 1978), but the one at Tc

is still not understood

Ferroelectrics – Physical Effects

166

Fig. 5. (c)

Temperature variations of thermal conductivity along three principal directions of

thiourea single crystal. (Menon & Philip, 2003)

well. Even below this transition temperature the material continues to remain ferroelectric.

Based on the measurement of the hydrostatic pressure dependence of the crystal structure of

DLP above and below the respective phase transitions, Gesi & Ozawa (1975) have proposed

that the phases above and below Tc

are isomorphous to each other. However, on the basis

of polarizing microscopic observations and dielectric constant measurements, Gesi (1984)

has concluded that the two phases above and below Tc

are not isostructural.

The crystal structure of DLP is tetragonal at room temperature (Ferroni & Orioli, 1959). The

lead atoms are located at 4a positions and calcium atoms at 8b positions. Studies on the

pyroelectric properties of DLP associated with its phase transitions have led to the

conclusion that DLP crystal between Tc

and Tc

is tetragonal and polar, the point group in

this phase being C

or C

4ν

(Osaka et al., 1975). Raman, infrared and dielectric properties of

this crystal has been studied by earlier workers (Nagae et al., 1976, Takashige et al., 1978).

The phase diagrams of mixed crystal system DSP-DLP, where DSP stands for Dicalcium

Strontium propionate, has been determined by Nagae et al. (1976) from dielectric and

dialatometric measurements. Nage et al., (1976) have reported Raman scattering spectra of

DSP and DLP between 73 and 423K. They concluded that both phase transitions of these

two materials are of the order – disorder type since no soft modes are observed, implying

that these transitions are most probably isomorphous. Takashige et al. (1978) have reported

the piezoelectric and elastic properties of ferroelectric DLP over a wide temperature region,

including the ferroelectric-paraelectric phase transition point (Tc

Even though the specific heat of DLP was reported way back in 1965 (Nakamura et al.,

1965), other thermal properties such as thermal conductivity were not. Moreover, systematic

thermal analysis following thermogravimetry or scanning calorimetry through Tc

and Tc

have not been reported. These measurements in DLP through the transition temperatures

have been reported by Manjusha & Philip (2008). These authors have reported thermal

Thermal Conduction Across Ferroelectric Phase Transitions: Results on Selected Systems

167

transport properties of the sample, thermal diffusivity, effusivity, conductivity and specific

heat capacity as a function of temperature following PPE technique. The anisotropy in

thermal diffusivity/conductivity along the principal axes as well as their variation through

these transition temperatures was measured.

The variations of thermal properties, shown in figures 6a and 6b, clearly indicate that the

thermal properties undergo anomalous variations during phase transitions at 191 K and 333

K. In general, the thermal diffusivity and thermal conductivity show an anomalous decrease

during transitions, whereas the heat capacity shows a corresponding anomalous increase.

Being an electrical insulator crystal, the major contribution to the heat capacity of DLP is

from lattice phonons and the electronic contribution to heat capacity is very small. As the

phonon modes undergo variations due to mode instability at the transition points, they

absorb excess energy giving rise to enhancement in heat capacity. This is found to get

reflected in the DSC curve as well. Again, during the transitions, the phonon mean-free path

increases, resulting in a decrease in thermal resistance or a corresponding increase in

thermal diffusivity and thermal conductivity. The anisotropy in thermal conductivity is not

very high for this crystal. The maximum thermal conduction occurs along the c-axis, which

is the direction of spontaneous polarization.

160 180 200 220 240 260 280 300 320 340 360

0.9

1.0

1.1

1.2

1.3

1.4

Temperature(K)

Thermal diffusivity (x10

-2

-1

)

6.38

6.40

6.42

6.44

6.46

6.48

6.50

6.52

6.54

Thermal effusivity (x10

-2

Jcm

-2

-1

-1/2

)

Fig. 6. (a) Variation of thermal diffusivity and thermal effusivity for DLP crystal, cut with

faces normal to the c-axis (Manjusha & Philip, 2008)

Many experimental and theoretical studies have been carried out by different workers to

understand the mechanisms of phase transitions in potassium selenate (K

SeO

) single

crystals, ever since the discovery of ferroelectricity and successive phase transitions in this

crystal (Aiki et al., 1969). With the occurrence of ferroelectric phase, this material undergoes

an incommensurate phase (IC phase) transition. Potassium selenate undergoes three

successive phase transitions at temperatures T

= 745 K, T

= 129.5 K and T

= 93 K (Aiki et al

1969). The crystal exhibits hexagonal structure in phase I, with space group D

46h

(P63/mmc)

(Shiozaki et al 1977), which changes to an orthorhombic structure (phase II) with space

Ferroelectrics – Physical Effects

168

160 180 200 220 240 260 280 300 320 340 360

0.31

0.32

0.33

0.34

0.35

0.36

0.37

0.38

0.39

Temperature(K)

Specific heat capacity (Jg

-1

)

0.60

0.62

0.64

0.66

0.68

0.70

0.72

0.74

0.76

Thermal conductivity (x10

-2

Wcm

-1

)

Fig. 6. (b) Variation of thermal conductivity and specific heat capacity for DLP crystal, cut

with faces normal to the c-axis (Manjusha & Philip, 2008)

group D

16 2h

(Pnam) at T

(Kalman et al 1970). Then, phase II changes into an incommensurate

one (phase III) at T

(Iizumi et al., 1977); this is a second order phase transition. It undergoes

an IC phase transition at T

, below which the crystal is commensurate and ferroelectric with

a small spontaneous polarization along the c direction.

Many experimental studies such as dielectric measurements (Aiki et al., 1969, Aiki et al.,

1970), X-ray and neutron diffraction (Iizumi et al., 1977; Ohama, 1974; Terauchi et al., 1975),

ESR (Aiki, 1970), Raman and Brillouin scattering (Wada et al., 1977a; Wada et al., 1977b;

Yagi et al., 1979), ultrasound velocity, attenuation and dispersion studies (Hoshizaki et al.,

1980; Shiozaki, 1977) etc have been reported near T

and T

. The variations in specific heat

capacity and thermal expansion of K

SeO

in the low temperature phase have also been

reported before (Aiki et al., 1970; Gupta et al., 1979). Thermal expansion along the c-axis

exhibits a discontinuity at the incommensurate to commensurate transition. Specific heat

measurements show anomalies at T

and T

, indicating that the transition at T

is second

order and that the one at T

is first order (Aiki et al., 1970). In spite of all these

measurements reported at temperatures T

and T

, only very few experimental results have

been reported near T

( Unruh et al., 1979; Inoue et al., 1979; Cho & Yagi, 1980; Gupta et al.,

1979) because of the inherent difficulties involved in carrying out precision experiments at

high temperatures. The variation of the specific heat capacity across the structural transition

at T

has not been reported so far for this material. More experimental data are still required

for a better understanding of the high temperature phase of this material.

The thermal diffusivity, thermal conductivity and heat capacity of K

SeO

as it goes through

the IC phase between 129.5 and 93 K have been measured by Philip & Manjusha (2009). The

anisotropy in thermal conductivity along the three principal directions of this crystal and its

variation with temperature are brought out and discussed by these authors. Differential

scanning calorimetric (DSC) measurements across the high temperature phases have been

carried out to determine anomalies in enthalpy during transition from phase I to phase II, and

Lallart M. (ed.) Ferroelectrics - Physical Effects

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